Volume 2, Number 1

Volume 2, Number 1, 1997

Closed Orbits and Chaotic Dynamics of a Charged Particle in a Periodic Electromagnetic Field

Abstract

We study motion of a charged particle on the two dimensional torus in a constant direction magnetic field. This analysis can be applied to the description of electron dynamics in metals, which admit a $2$-dimensional translation group (Bravais crystal lattice). We found the threshold magnetic value, starting from which there exist three closed Larmor orbits of a given energy. We demonstrate that if there are n lattice atoms in a primitive Bravais cell then there are $4+n$ different Larmor orbits in the nondegenerate case. If the magnetic field is absent the electron dynamics turns out to be chaotic, dynamical systems on the corresponding energy shells possess positive entropy in the case that the total energy is positive.

Using two classical integrable problems, we demonstrate some methods of a new theory of orbital classification for integrable Hamiltonian systems with two degrees of freedom. We show that the Liouville foliations (i.e., decompositions of the phase space into Liouville tori) of the two systems under consideration are diffeomorphic. Moreover, these systems are orbitally topologically equivalent, but this equivalence cannot be made smooth.

Mel'nikov's perturbation method for showing the existence of transversal intersections between invariant manifolds of saddle fixed points of dynamical systems is extended here to second order in a small parameter $\epsilon$. More specifically, we follow an approach due to Wiggins and derive a formula for the second order Mel'nikov vector of a class of periodically perturbed $n$-degree of freedom Hamiltonian systems. Based on the simple zero of this vector, we prove an $O(\epsilon^2)$ sufficient condition for the existence of isolated homoclinic (or heteroclinic) orbits, in the case that the first order Mel'nikov vector vanishes identically. Our result is applied to a damped, periodically driven $1$-degree-of-freedom Hamiltonian and good agreement is obtained between theory and experiment, concerning the threshold of heteroclinic tangency.

Nonintegrability of Hamiltonian systems in vortex dynamics. I. The interactions of three vortex rings

Abstract

The problem of the interaction of three coaxial vortex rings in an ideal fluid is studied. By introducing an artifical small parameter and using the separatrix split method the nonintegrability of the restricted problem is proved analytically.

Three-parametrical family of systems with two degrees of freedom of a kind
$$\begin{array}{rcl}
\dfrac{d^2x_1}{dt^2}+x_1&=&-2\varepsilon x_1x_2\\
\dfrac{d^2x_2}{dt^2}+x_2-x_2^2&=&\varepsilon (-x_1^2+\delta\dot{x}_2+\gamma x_2\dot{x}_2),
\end{array}\qquad\qquad(*)$$
where $\varepsilon>0$ is considered. Analytical research of trajectories behaviour of the system $(*)$ is carried out when $\varepsilon$ is small.
The given research is connected, first of all, to the analysis of resonant zones.
Alongside with the initial system, another system
$$\ddot{x}_2+x_2-x_2^2=\varepsilon(-A^2\sin^2t+\delta\dot{x}_2+\gamma x_2\dot{x}_2)\qquad\qquad(**)$$
that is "close" to the original, is considered. A good concurrence of results for Poincare mapping, induced by an equation $(**)$ when $\delta=\gamma=0$, and for the mapping that was constructed by Henon and Heiles, is established. In addition, for system $(*)$ a transition to nonregular dynamics is numerically analyzed at increase of parameter $\varepsilon$ and $\delta=\gamma=0$. It is established, that the transition to nonregular dynamics is connected, in particular, with the period doubling bifurcation (known as Feigenbaum's script), and $\varepsilon_\infty\approx0.95$.

We consider a new class of Hamiltonian dynamical systems with two degrees of freedom whose kinetic energy is a function of momenta's modules. Equations of motion for such systems are easily integrated in each of succesive time intervals; thus, in principle, all the trajectories can be found explicitly. A Poincare mapping for such systems can be reduced to a mapping of the least positive root for a system of transcendental (in the general case) equations.

On the other hand, dynamical systems of this type exhibit a number of properties typical to non-intergable systems (e.g., an existence of stable and unstable periodic orbits, their bifurcations, creation of stochastic layers in vicinity of destroyed separatrices, regions of global chaotic motion, etc.).

As an example, a system with a simple potential that is quadratic in both coordinates is studied.

Taking a classical problem of motion of a rigid body in a gravitational field as an example, we consider Feigenbaum's script for transition to stochasticity. Numerical results are obtained using Andoyer-Deprit's canonical variables. We calculate universal constants describing "doubling tree" self-duplication scaling. These constants are equal for all dynamical systems, which can be reduced to the study of area-preserving mappings of a plan onto itself. We show that stochasticity in Euler-Poisson equations can progress according to Feigenbaum's script under some restrictions on the parameters of our system.

Quadratically Integrable Geodesic Flows on the Torus and on the Klein Bottle

Abstract

We classify integrable geodesic flows on the torus and on the Klein bottle, which in addition admit a quadratic in momenta first integral. We also study cases if there exists an additional linear integral.

We show that an invariant surface allows to construct the Jacobi vector field along a geodesic line and construct the formula for the normal part of the Jacobi field. If a geodesic line is the transversal intersection of two invariant surfaces (such situation we have, for example, if the geodesic line is hyperbolic) than we can construct the fundamental solution of Jacobi equation $\ddot{u}=-K(t)u$. That was done for quadratically integrable geodesic flows.